Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-9q^3 + 144q^2 - 567q}{-9q^2 + 72q - 63}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-9q(q^2 - 16q + 63)} {-9(q^2 - 8q + 7)} $ $ p = \dfrac{9q}{9} \cdot \dfrac{q^2 - 16q + 63}{q^2 - 8q + 7} $ Simplify: $ p = q \cdot \dfrac{q^2 - 16q + 63}{q^2 - 8q + 7}$ Next factor the numerator and denominator. $ p = q \cdot \dfrac{(q - 7)(q - 9)}{(q - 7)(q - 1)}$ Assuming $q \neq 7$ , we can cancel the $q - 7$ $ p = q \cdot \dfrac{q - 9}{q - 1}$ Therefore: $ p = \dfrac{ q(q - 9)}{ q - 1 }$, $q \neq 7$